Biharmonic equation

In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. It is written as

or

or

where is the fourth power of the del operator and the square of the laplacian operator (or ), and it is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in dimensions as: